The purpose of this article is to establish a Wiener-type sufficient condition for the regularity
of a boundary point \(x_{0}\in \partial \Omega \)  for bounded solutions of Eq.
(REF ) under assumptions (\({\rm g}_{0}\) )–(\({\rm g}_{3}\) ).
Boundary regularity in terms of the Sobolev variational capacity is a classical topic in the
contemporary theory of PDE, which goes back to
Lebesgue [1]} and Wiener [2]}, [3]}.
The papers
[4]}, [5]}, [6]}, [7]},
[8]}, [9]}
provided the basic contribution to the boundary regularity for
quasilinear elliptic equations with standard growth conditions.
For the variable \(p(x)\) -growth, we cite [10]} in the log-case,
and [11]} for non-logarithmic conditions
(REF ), (REF ) on the exponent \(p(\cdot )\) .
In the Orlicz case, the Wiener criterion for the regularity of Sobolev boundary point
was established by Ki-Ahm Lee and Se-Chan Lee [12]}.
Under generalized Orlicz growth and logarithmic counterparts of (REF ),
Harjulehto and Hästö [13]} obtained the following capacity density
condition for the regularity
of a boundary point \(x_{0}\in \partial \Omega \) : there exists \(c\in (0,1)\)  and \(R>0\)  such that
\(\mathrm {Cap}_{G}\big (B_{\rho }(x_{0})\setminus \Omega , B_{2\rho }(x_{0})\big )\geqslant c\,\mathrm {Cap}_{G}\big (B_{\rho }(x_{0}), B_{2\rho }(x_{0})\big )\) 
for all \(0<\rho <R\) , \(G(\cdot , {\rm v})\asymp g(\cdot , {\rm v}){\rm v}\) .
This result was improved by Benyaiche and Khlifi [14]} to a full-fledged
Wiener-type criterion: the point \(x_{0}\)  is regular if and only if for some \(R>0\) ,
\(\int _{0}^{R}g^{-1}_{x_{0}}\bigg ( \frac{{\rm Cap}_{G}\big (B_{\rho }(x_{0})\setminus \Omega , B_{2\rho }(x_{0}) \big )}{\rho ^{n-1}}\bigg )\,d\rho =+\infty \) 
